Remark. It can be shown that such cardinal $\kappa$ belongs to the half-interval $[\mathfrak b,\mathfrak d)$, where $\mathfrak b$ and $\mathfrak d$ are the boundedness and dominating numbers of the poset $\omega^\omega$.

Therefore, under $\mathfrak b=\mathfrak d$ the answer to the above problem is "No".

Is the same "No" true in ZFC?

This question can be reformulated in topological terms as follows. Following Cascales, Orihuela and Tkachuk, we say that a topological space $X$ is strongly $\omega^\omega$-dominated if there exists a family $(K_\alpha)_{\alpha\in\omega^\omega}$ of compact subsets of $X$ such that $K_\alpha\subset K_\beta$ for every $\alpha\le\beta$ in $\omega^\omega$ and for every compact set $K\subset X$ there exists $\alpha\in\omega^\omega$ such that $K\subset K_\alpha$.

It is easy to see that Problem 1 is equivalent to the topological

Problem 2. Is the space $\kappa^\kappa$ strongly $\omega^\omega$-dominated for some uncountable cardinal $\kappa$?

Here $\kappa^\kappa$ carries the Tychonoff product topology and $\kappa$ is endowed with the standard order topology.

A Known Related Fact (Cascales-Orihuela-Tkachuk). For any uncountable cardinal $\kappa$ the space $\omega^\kappa$ is not strongly $\omega^\omega$-dominated and hence does not admit a monotone cofinal map $f:\omega^\omega\to\omega^\kappa$.

Concerning the consistency strength of the statements in Problems 1,2, let us recall the following result of Cummings and Shelah. For a cardinal $\kappa$ let $\mathfrak b_\kappa$ and $\mathfrak d_\kappa$ be the boundedness and dominating numbers of $\kappa^\kappa$ endowed with the partial preorder $f\le^* g$ iff $|\{\alpha\in\kappa: f(\alpha)>g(\alpha)\}|<\kappa$.

Theorem. Let $F$ be a class function assigning to each regular cardinal $\kappa$ a triple of cardinals $(\beta(\kappa),\delta(\kappa),\lambda(\kappa))$ such that
$cf(\lambda(\kappa))>\kappa<\beta(\kappa)=cf(\beta(\kappa))\le\delta(\kappa)\le\lambda(\kappa)$ and $\lambda(\kappa)\le\lambda(\kappa')$ for any regular cardinals $\kappa<\kappa'$. Then there exists a class forcing poset $\mathbb P$ preserving all cardinals and cofinalities such that in the generic extension, $\mathfrak b_\kappa=\beta(\kappa)$, $\mathfrak d_\kappa=\delta(\kappa)$ and $|\kappa^\kappa|=\lambda(\kappa)$.

This result of Cumming and Shelah implies the consistency of $\mathfrak b=\omega_1$, $\mathfrak d=\omega_2$ and $\mathfrak b_\kappa=\mathfrak d_\kappa>\omega_2$ for all regular uncountable cardinals $\kappa$.

This impies that the consistency of the answer "No" to Problems 1,2 is strictly weaker that the consistency of the equality $\mathfrak b=\mathfrak d$.

On the other hand, the same result of Cummings and Shelah implies the consistency of $\mathfrak b=\omega_1$ and $\mathfrak b_{\omega_1}=\mathfrak d_{\omega_1}=\mathfrak d=\omega_1^{\omega_1}=\omega_2$, witnessing that a (desirable for me negative answer) to Problem 1 should use hardly the monotonicity (not only cofinality) of the map $f:\omega^\omega\to\kappa^\kappa$.

1 Answer
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My former doctoral student Lubomyr Zdomskyy has resolved this problem, noticing that adding $\omega_2$ Cohen reals to a model of GCH produces a model in which the cardinal $\omega_1$ admits a monotone cofinal map $\omega^\omega\to(\omega_1)^{\omega_1}$.

Alternatively the same result can be derived from the existence of a K-Lusin set of cardinality $\mathfrak c=2^{\omega_1}$ in $\omega^\omega$.

A subset $L$ of a Polish space $X$ is called a K-Lusin set in $X$ if for every compact subset $K\subset X$ the intersection $K\cap L$ is countable.

Theorem (Zdomskyy). If there exists a $K$-Lusin set $L\subset\omega^\omega$ of cardinality $|L|=\mathfrak c=2^{\omega_1}$, then there exists a mootone cofinal map $f:\omega^\omega\to(\omega_1)^{\omega_1}$.

Proof. Let $g:L\to (\omega_1)^{\omega_1}$ be any surjective map. Observe that for every $x\in\omega^\omega$ the set $L_x=\{y\in L:y\le x\}$ is countable. Then the formula $f(x)=\sup(g(L_x))$ defines a required cofinal monotone map $f:\omega^\omega\to(\omega_1)^{\omega_1}$. $\square$

$\begingroup$Brilliant solution. It is a very old fact that there are Luzin sets in the Cohen model. (Briefly, every Borel meager set is coded using countably many Cohen reals, and all other Cohen reals avoid it.) Every Luzin set is K-Luzin. Thus, I believe that the existence of K-Luzin sets in the Cohen model should be dated to the beginning of the theory of forcing. (It would be nice if someone can add here a more precise dating/source.)$\endgroup$
– Boaz TsabanJul 15 '16 at 6:55